L8a15

L8a14

L8a16

	Visit L8a15's page at Knotilus! Visit L8a15's page at the original Knot Atlas!
	L8a15 is $8_{3}^{3}$ in the Rolfsen table of links.

L8a15 Further Notes and Views

Knot presentations

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X16,12,9,11 X12,16,13,15 X2536 X4,9,1,10
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -4}, {8, -2, 5, -6, 4, -3, 6, -5}

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw-2uv-2uw+2u-2vw+2v+2w-1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{-7}-q^{-6}+4q^{-5}-4q^{-4}+6q^{-3}-4q^{-2}+q+4q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+3a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{4}-a^{2}z^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-3za^{7}+2a^{7}z^{-1}+z^{6}a^{6}+2z^{4}a^{6}-6z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{7}a^{5}+2z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+4z^{6}a^{4}-5z^{4}a^{4}-a^{4}z^{-2}+3a^{4}+z^{7}a^{3}+2z^{5}a^{3}-3z^{3}a^{3}+3z^{6}a^{2}-5z^{4}a^{2}+2z^{2}a^{2}+3z^{5}a-5z^{3}a+z^{4}-z^{2}$ (db)

Vassiliev invariants

V2 and V3:

(0, ${\frac {27}{2}}$)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:L8a15/V 2,1 Data:L8a15/V 3,1 Data:L8a15/V 4,1 Data:L8a15/V 4,2 Data:L8a15/V 4,3 Data:L8a15/V 5,1 Data:L8a15/V 5,2 Data:L8a15/V 5,3 Data:L8a15/V 5,4 Data:L8a15/V 6,1 Data:L8a15/V 6,2 Data:L8a15/V 6,3 Data:L8a15/V 6,4 Data:L8a15/V 6,5 Data:L8a15/V 6,6 Data:L8a15/V 6,7 Data:L8a15/V 6,8 Data:L8a15/V 6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of L8a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1               2   -2 -1             2 1   1 -3           3 3     0 -5         3 1       2 -7       1 3         2 -9     3 3           0 -11   1 4             3 -13                   0 -15 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Link[8, Alternating, 15]]
Out[2]=	8
In[3]:=	PD[Link[8, Alternating, 15]]
Out[3]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], X[16, 12, 9, 11], X[12, 16, 13, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=	GaussCode[Link[8, Alternating, 15]]
Out[4]=	GaussCode[{1, -7, 2, -8}, {7, -1, 3, -4}, {8, -2, 5, -6, 4, -3, 6, -5}]
In[5]:=	BR[Link[8, Alternating, 15]]
Out[5]=	BR[Link[8, Alternating, 15]]
In[6]:=	alex = Alexander[Link[8, Alternating, 15]][t]
Out[6]=	ComplexInfinity
In[7]:=	Conway[Link[8, Alternating, 15]][z]
Out[7]=	ComplexInfinity
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[Link[8, Alternating, 15]], KnotSignature[Link[8, Alternating, 15]]}
Out[9]=	{Infinity, -2}
In[10]:=	J=Jones[Link[8, Alternating, 15]][q]
Out[10]=	-7 -6 4 4 6 4 4 -3 + q - q + -- - -- + -- - -- + - + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[Link[8, Alternating, 15]][q]
Out[12]=	-24 3 3 4 6 3 3 2 -8 2 -4 -1 + q + --- + --- + --- + --- + --- + --- + --- + q + -- - q + 22 20 18 16 14 12 10 6 q q q q q q q q -2 2 4 q - q + q
In[13]:=	Kauffman[Link[8, Alternating, 15]][a, z]
Out[13]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z - 2 2 2 z z z z z 2 2 2 6 2 8 2 3 3 3 5 3 4 z + 2 a z - 6 a z - 3 a z - 5 a z - 3 a z + 2 a z + z - 2 4 4 4 6 4 8 4 5 3 5 7 5 5 a z - 5 a z + 2 a z + a z + 3 a z + 2 a z + a z + 2 6 4 6 6 6 3 7 5 7 3 a z + 4 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Link[8, Alternating, 15]], Vassiliev[3][Link[8, Alternating, 15]]}
Out[14]=	27 {0, --} 2
In[15]:=	Kh[Link[8, Alternating, 15]][q, t]
Out[15]=	3 2 1 1 4 3 3 1 3 -- + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t 3 1 3 t 3 2 ----- + ---- + ---- + - + 2 q t + q t 5 2 5 3 q q t q t q t